IMO 1985 Shortlist
IMO 1985 Shortlist — 22 problems. 22 problems.
IMO 1985 Shortlist
22 problems · Source: IMO Compendium
Problems
| # | Origin | Problem |
|---|---|---|
| 1 | MON | Given a set M of 1985 positive integers, none of which |
| 2 | BRA | A polyhedron has 12 faces and is such that: |
| 3 | NET | The weight w(p) of a polynomial p, p(x) = n |
| 4 | AUS | Each of the numbers in the set N = {1, 2, 3, . . ., n −1}, |
| 5 | ROM | Let D be the interior of the circle C and let A \inC. Show |
| 6 | POL | Let xn = |
| 7 | — | 1a.(CZS 3) The positive integers x1, . . . , xn, n \geq3, satisfy x1 < x2 < |
| 8 | — | 1b.(TUR 5) Find the smallest positive integer n such that |
| 9 | — | 2a.(USA 3) Determine the radius of a sphere S that passes through the |
| 10 | — | 2b.(VIE 1) |
| 11 | — | 3a.(USS 3) Find a method by which one can compute the coefficients |
| 12 | — | 3b.(GBR 4) A sequence of polynomials Pm(x, y, z), m = 0, 1, 2, . . ., in |
| 13 | — | 4a.(BUL 1) |
| 14 | — | 4b.(IRE 4) A set of 1985 points is distributed around the circumference |
| 15 | — | 5a.(FRA 3) Let K and K′ be two squares in the same plane, their sides |
| 16 | — | 5b.(BEL 2) |
| 17 | — | 6a.(SWE 3)IMO6 The sequence f1, f2, . . . , fn, . . . of functions is defined |
| 18 | — | 6b.(CAN 5) Let x1, x2, . . . , xn be positive numbers. Prove that |
| 19 | ISR | For which integers n \geq3 does there exist a regular n-gon in the |
| 20 | GBR | A circle whose center is on the side ED of the cyclic |
| 21 | IRE | The tangents at B and C to the circumcircle of the acute-angled |
| 22 | USS | A circle with center O passes through points A and C and |